In this article, we present a numerical iterative method for the solution of internal viscous and incompressible flows in real porous three-dimensional bodies at their pore scale. We use the penalized formulation of the problem involving velocity and vorticity: an operator splitting allows to split apart the diffusion (inherited from Stokes equation) and the penalization phenomena (which takes into account the solid matrix). By means of the numerical analysis of the splitting, we exhibit the penalization coefficient which is actually effective. This method allows to deal only with fast-evaluation operators, that is to say scaling at most as $\mathcal{O}(n\log n)$ where n is the number of underlying grid points, such as straightforward computations of finite differences schemes or FFT solver. The numerical analysis and implementation solutions are presented, and validated on various digital rock physics geometries acquired by micro-tomography, using numerical and physical diagnostics. To enforce this validation, we also present permeability estimations of several porous samples. The simulation of transport of passive and active scalars is finally investigated in order to perform the practical upscaling to 1D models of transport and diffusion at the Darcy scale.

在本文中，我们提出了一种数值迭代方法，用于求解内部多孔和不可压缩流动在实际多孔三维物体中的孔尺度。我们使用涉及速度和涡度的问题的惩罚公式：算符拆分允许拆分扩散（从Stokes方程继承）和惩罚现象（考虑了固体矩阵）。通过对分裂的数值分析，我们展示了实际上有效的惩罚系数。这种方法只允许处理快速求值运算符，也就是说最多只能扩展为$\mathcal{Ø}（ñ\mathrm{日志}ñ）$其中n是基础网格点的数量，例如有限差分方案的简单计算或FFT求解器。提出了数值分析和实现解决方案，并使用数值和物理诊断方法对通过显微断层摄影术获得的各种数字岩石物理几何形状进行了验证。为了执行此验证，我们还提出了几个多孔样品的渗透率估计。最后研究了无源和有源标量传输的仿真，以便在达西尺度上将传输和扩散的一维模型实际升级到一维模型。